Fourier transforms and Ringel–Hall algebras of valued quivers

Abstract

In this paper, we follow an idea of Lusztig to define the Fourier transform on the Ringel–Hall algebra of a valued quiver (given by a quiver with automorphism). As an application, this provides a direct proof of the fact that the Ringel–Hall algebra of a valued quiver is independent of its orientation. Furthermore, by combining the BGP-reflection operators defined on double Ringel–Hall algebras of valued quivers with Fourier transforms, we obtain an alternative construction of Lusztig’s symmetries of the associated quantum enveloping algebras. This generalizes a result of Sevenhant and Van den Bergh in the quiver case.